Monday, May 2, 2016

An attempt to refute my Big-Bang singularity solution

I learned recently about a paper which attempts to refute one of my papers. While being sure about my proofs, I confess that I was a bit worried, you never know when you made a mistake, a silly assumption that you overlooked. But as I was reading the refutation paper, my worries dissipated, and were replaced by amusement and I actually had a lot of fun. Because that so-called refutation was something like: "I will refute Pythagoras's Theorem by showing that it doesn't apply to triangles that are not right."

My paper in cause about Big-Bang singularities is arXiv:1112.4508 (The Friedmann-Lemaitre-Robertson-Walker Big Bang singularities are well behaved). As it is known, the main mathematical tool used in General Relativity is semi-Riemannian geometry, and this works only as long as the metric is regular. The metric ceases to be regular at singularities, but I developed the extension of semi-Riemannian geometry at some degenerate metrics, so it applies to a large class of singularities, in arxiv:1105.0201. And this allowed me to find descriptions of such singularities in terms of quantities that are still invariant, but as opposed to the usual ones, they remain finite at singularities. More about this can be found in my PhD thesis arxiv:1301.2231. In the paper arXiv:1112.4508, I give a theorem that shows that, if the scaling function of the FLRW universe is smooth at the Big-Bang singularity, then I can apply the tools I developed previously, and get a finite description of both the geometry, and the physical quantities involved.

The paper attempting to refute my result is arxiv:1603.02837 (Behavior of Friedmann-Lemaitre-Robertson-Walker Singularities, by L. Fernández-Jambrina). Both my paper and this one appeared this year in International Journal of Theoretical Physics. I think F-J is a good researcher and expert in singularities. But for some reason, he didn't like my paper, and he "refuted" it. The "refutation" simply takes the case that was explicitly not covered in my theorem, namely when the scaling function of the FLRW solution is not derivable at the singularity, and checks that indeed my tools don't work in this case. Now, while my result is much more humble than Pythagoras's Theorem, I will use it for comparison, since it is well-known by everybody. You can't refute Pythagoras's Theorem by taking triangles that are not right, and proving that the sum of squares of two sides is different than the square of the third. Simply because the Theorem makes clear in its hypothesis that it refers only to right triangles. My theorem also states clearly that the result doesn't refer to FLRW models whose scaling function is not derivable at the singularity. And F-J even copies the Theorem's enounce in his paper, so how could he miss this? So what F-J said is that my theorem can't be applied to some cases, which I made clear that I leave out (I don't claim my theorem solves everything, neither that it cures cancer). Now, is the case when the scaling function is not derivable important? Yes, at least historically, because some classical solutions fit here. But the cases covered by my theorem include what we know today about inflation. So I think that my result is not only correct, but also significant. In addition to this, F-J says that I actually don't remove the Big-Bang singularity. This is also true, and stated in my paper from the beginning. I don't remove the singularities, I just try to understand them to describe them in terms of finite quantities that make sense both geometrically and physically. But he wrote it as if I claim that I try to remove them and he proves that I don't, not that I accept them and provide a finite-quantities description of them.